rayoptics.elem.profiles module

Module for different surface profile shapes

The profiles module captures the geometric shape aspect of the optical interface. The SurfaceProfile base class specifies an api that subclasses implement to provide different shapes. It also provides generic implementations of ray-profile intersections.

resize_list(lst, new_length, null_item=None)

intersect_parabola(cv, p, d, z_dir=1.0)

Intersect a parabolid, starting from an arbitrary point.

Parameters:

• cv – vertex curvature

• p – start point of the ray in the profile’s coordinate system

• d – direction cosine of the ray in the profile’s coordinate system

• z_dir – +1 if propagation positive direction, -1 if otherwise

class SurfaceProfile

Bases: object

Base class for surface profiles.

update()

f(p)

Returns the value of the profile surface function at point \(\boldsymbol{p}\).

\(f({\boldsymbol{p}}) = 0\)

where \(\boldsymbol{p}=({x, y, z})^T\)

df(p)

Returns the gradient of the profile surface function at point \(\boldsymbol{p}\).

\(df(\boldsymbol{p}) = \partial{f(\boldsymbol{p})}/\partial{\boldsymbol{p}}\)

normal(p)

Returns the unit normal of the profile at point \(\boldsymbol{p}\).

sag(x, y)

Returns the sagitta (z coordinate) of the surface at x, y.

profile(sd, dir=1, steps=6)

Return a 2d polyline approximating the surface profile.

Parameters:

• sd – semi-diameter of the profile

• dir – +1 for profile from neg to postive direction, -1 if otherwise

• steps – number of points to generate

apply_scale_factor(scale_factor)

Apply scale_factor to the profile definition.

intersect(p0, d, eps, z_dir)

Intersect a profile, starting from an arbitrary point.

Parameters:

• p0 – start point of the ray in the profile’s coordinate system

• d – direction cosine of the ray in the profile’s coordinate system

• z_dir – +1 if propagation positive direction, -1 if otherwise

• eps – numeric tolerance for convergence of any iterative procedure

Returns:

distance to intersection point s1, intersection point p

Return type:

tuple

Raises:

TraceMissedSurfaceError –

intersect_welford(p, d, eps, z_dir)

Intersect a profile, starting from an arbitrary point.

From Welford, Aberrations of Optical Systems (ISBN-10: 0852745648), eqs 4.34 thru 4.41.

Parameters:

• p0 – start point of the ray in the profile’s coordinate system

• d – direction cosine of the ray in the profile’s coordinate system

• z_dir – +1 if propagation positive direction, -1 if otherwise

• eps – numeric tolerance for convergence of any iterative procedure

Returns:

distance to intersection point s1, intersection point p

Return type:

tuple

Raises:

TraceMissedSurfaceError –

intersect_spencer(p0, d, eps, z_dir)

Intersect a profile, starting from an arbitrary point.

From Spencer and Murty, General Ray-Tracing Procedure

Parameters:

• p0 – start point of the ray in the profile’s coordinate system

• d – direction cosine of the ray in the profile’s coordinate system

• z_dir – +1 if propagation positive direction, -1 if otherwise

• eps – numeric tolerance for convergence of any iterative procedure

Returns:

distance to intersection point s1, intersection point p

Return type:

tuple

Raises:

TraceMissedSurfaceError –

intersect_scipy(p0, d, eps, z_dir)

Intersect a profile, starting from an arbitrary point.

Uses the newton method of scipy.optimize.root_scalar.

Parameters:

• p0 – start point of the ray in the profile’s coordinate system

• d – direction cosine of the ray in the profile’s coordinate system

• z_dir – +1 if propagation positive direction, -1 if otherwise

• eps – numeric tolerance for convergence of any iterative procedure

Returns:

distance to intersection point s1, intersection point p

Return type:

tuple

Raises:

TraceMissedSurfaceError –

class Spherical(c=0.0, r=None)

Bases: SurfaceProfile

Spherical surface profile parameterized by curvature.

The sag \(z\) is given by:

\(z = R - \sqrt{R^2 - x^2 - y^2}\)

where \(R = 1/c\)

—

__init__(c=0.0, r=None)

initialize a Spherical profile.

Parameters:

• c – curvature

• r – radius of curvature. If zero, taken as planar. If r is specified, it overrides any input for c (curvature).

property r

listobj_str()

copyFrom(other)

copyDataFrom(other)

mutate(new_profile)

apply_scale_factor(scale_factor)

Apply scale_factor to the profile definition.

flip()

intersect_tangent_plane(p, d, eps, z_dir)

intersect(p, d, eps, z_dir)

Intersection with a sphere, starting from an arbitrary point.

f(p)

surface function for Spherical profile

This function implements Spencer’s eq 25, with kappa=1 (i.e. spherical).

To see this, start with the code: F = p[2] - 0.5*cv*(np.dot(p, p))

Expand np.dot(p, p): F = p[2] - 0.5*cv*(p[0]*p[0] + p[1]*p[1] + p[2]*p[2])

in Spencer’s notation: rho**2 = p[0]*p[0] + p[1]*p[1] Z = p[2]

Substituting notation, the result is: F = Z - 0.5*cv*(rho**2 + Z**2)

which is Spencer’s eq 25.

df(p)

Returns the gradient of the profile surface function at point \(\boldsymbol{p}\).

\(df(\boldsymbol{p}) = \partial{f(\boldsymbol{p})}/\partial{\boldsymbol{p}}\)

sag(x, y)

Returns the sagitta (z coordinate) of the surface at x, y.

profile(sd, dir=1, steps=6)

Generate a profile curve for the segment sd.

class Conic(c=0.0, cc=0.0, r=None, ec=None)

Bases: SurfaceProfile

Conic surface profile parameterized by curvature and conic constant.

Conics produced for conic constant values:

• cc > 0.0: oblate spheroid

• cc = 0.0: sphere

• cc < 0.0 and > -1.0: ellipsoid

• cc = -1.0: paraboloid

• cc < -1.0: hyperboloid

Conics produced for conic asphere values:

• ec > 1.0: oblate spheroid

• ec = 1.0: sphere

• ec > 0.0 and < 1.0: ellipsoid

• ec = 0.0: paraboloid

• ec < 0.0: hyperboloid

The conic constant is related to the conic asphere as:

• cc = ec - 1

The sag \(z\) is given by:

\(z(r)=\dfrac{cr^2}{1+\sqrt[](1-\textbf{ec } c^2 r^2)}\)

where \(r^2 = x^2+y^2\)

—

__init__(c=0.0, cc=0.0, r=None, ec=None)

initialize a Conic profile.

Parameters:

• c – curvature

• r – radius of curvature. If zero, taken as planar. If r is specified, it overrides any input for c (curvature).

• cc – conic constant

• ec – conic asphere (= cc + 1). If ec is specified, it overrides any input for the conic constant (cc).

property r

property ec

listobj_str()

copyFrom(other)

copyDataFrom(other)

apply_scale_factor(scale_factor)

Apply scale_factor to the profile definition.

flip()

intersect_tangent_plane(p, d, eps, z_dir)

intersect(p, d, eps, z_dir)

Intersection with a conic, starting from an arbitrary point.

f(p)

surface function for Conic profile

This function implements Spencer’s eq 25, with kappa = ec = 1 + cc

df(p)

Returns the gradient of the profile surface function at point \(\boldsymbol{p}\).

\(df(\boldsymbol{p}) = \partial{f(\boldsymbol{p})}/\partial{\boldsymbol{p}}\)

sag(x, y)

Returns the sagitta (z coordinate) of the surface at x, y.

profile(sd, dir=1, steps=6)

Return a 2d polyline approximating the surface profile.

Parameters:

• sd – semi-diameter of the profile

• dir – +1 for profile from neg to postive direction, -1 if otherwise

• steps – number of points to generate

append_pt_to_2d_profile(surface_profile, y, poly_profile)

calc surface sag at y and append to poly if ok, else return None

aspheric_profile(surface_profile, sd, dir=1, steps=21)

class EvenPolynomial(c=0.0, cc=0.0, r=None, ec=None, coefs=None)

Bases: SurfaceProfile

Even Polynomial asphere, even terms up to 20th order, on base conic.

Conics produced for conic constant values:

• cc > 0.0: oblate spheroid

• cc = 0.0: sphere

• cc < 0.0 and > -1.0: ellipsoid

• cc = -1.0: paraboloid

• cc < -1.0: hyperboloid

Conics produced for conic asphere values:

• ec > 1.0: oblate spheroid

• ec = 1.0: sphere

• ec > 0.0 and < 1.0: ellipsoid

• ec = 0.0: paraboloid

• ec < 0.0: hyperboloid

The conic constant is related to the conic asphere as:

• cc = ec - 1

The sag \(z\) is given by:

\(z(r)=\dfrac{cr^2}{1+\sqrt[](1-\textbf{ec } c^2 r^2)}+\sum_{i=1}^{20} a_ir^{2i}\)

where \(r^2=x^2+y^2\)

—

initial_size = 10

__init__(c=0.0, cc=0.0, r=None, ec=None, coefs=None)

initialize a EvenPolynomial profile.

Parameters:

• c – curvature

• r – radius of curvature. If zero, taken as planar. If r is specified, it overrides any input for c (curvature).

• cc – conic constant

• ec – conic asphere (= cc + 1). If ec is specified, it overrides any input for the conic constant (cc).

• coefs – a list of even power coefficents, starting with the quadratic term, and not exceeding the 20th order term.

property r

property ec

get_by_order(order: int) → float

Access to coefficients via polynomial order

set_by_order(order: int, value: float)

Access to coefficients via polynomial order

listobj_str()

copyFrom(other)

copyDataFrom(other)

calc_max_nonzero_coef()

apply_scale_factor(scale_factor)

Apply scale_factor to the profile definition.

flip()

update()

sag(x, y)

Returns the sagitta (z coordinate) of the surface at x, y.

f(p)

Returns the value of the profile surface function at point \(\boldsymbol{p}\).

\(f({\boldsymbol{p}}) = 0\)

where \(\boldsymbol{p}=({x, y, z})^T\)

df(p)

Returns the gradient of the profile surface function at point \(\boldsymbol{p}\).

\(df(\boldsymbol{p}) = \partial{f(\boldsymbol{p})}/\partial{\boldsymbol{p}}\)

profile(sd, dir=1, steps=21)

Return a 2d polyline approximating the surface profile.

Parameters:

• sd – semi-diameter of the profile

• dir – +1 for profile from neg to postive direction, -1 if otherwise

• steps – number of points to generate

class RadialPolynomial(c=0.0, cc=None, r=None, ec=1.0, coefs=None)

Bases: SurfaceProfile

Radial Polynomial asphere, both even and odd terms, on base conic.

Conics produced for conic constant values:

• cc > 0.0: oblate spheroid

• cc = 0.0: sphere

• cc < 0.0 and > -1.0: ellipsoid

• cc = -1.0: paraboloid

• cc < -1.0: hyperboloid

Conics produced for conic asphere values:

• ec > 1.0: oblate spheroid

• ec = 1.0: sphere

• ec > 0.0 and < 1.0: ellipsoid

• ec = 0.0: paraboloid

• ec < 0.0: hyperboloid

The conic constant is related to the conic asphere as:

• cc = ec - 1

The sag \(z\) is given by:

\(z(r)=\dfrac{cr^2}{1+\sqrt[](1-\textbf{ec } c^2 r^2)}+\sum_{i=1}^{10} a_ir^i\)

where \(r^2 = x^2+y^2\)

—

initial_size = 10

__init__(c=0.0, cc=None, r=None, ec=1.0, coefs=None)

initialize a RadialPolynomial profile.

Parameters:

• c – curvature

• r – radius of curvature. If zero, taken as planar. If r is specified, it overrides any input for c (curvature).

• ec – conic asphere.

• cc – conic constant (= ec - 1). If cc is specified, it overrides any input for the conic asphere (ec).

• coefs – a list of radial coefficents, starting with the linear term, and not exceeding the 10th order term.

property r

property cc

get_coef(exp)

set_coef(exp, value)

get_by_order(order: int) → float

Access to coefficients via polynomial order

set_by_order(order: int, value: float)

Access to coefficients via polynomial order

listobj_str()

copyFrom(other)

copyDataFrom(other)

calc_max_nonzero_coef()

apply_scale_factor(scale_factor)

Apply scale_factor to the profile definition.

flip()

update()

sag(x, y)

Returns the sagitta (z coordinate) of the surface at x, y.

f(p)

Returns the value of the profile surface function at point \(\boldsymbol{p}\).

\(f({\boldsymbol{p}}) = 0\)

where \(\boldsymbol{p}=({x, y, z})^T\)

df(p)

Returns the gradient of the profile surface function at point \(\boldsymbol{p}\).

\(df(\boldsymbol{p}) = \partial{f(\boldsymbol{p})}/\partial{\boldsymbol{p}}\)

profile(sd, dir=1, steps=21)

Return a 2d polyline approximating the surface profile.

Parameters:

• sd – semi-diameter of the profile

• dir – +1 for profile from neg to postive direction, -1 if otherwise

• steps – number of points to generate

class YToroid(c=0.0, cR=0, cc=0.0, r=None, rR=None, ec=None, coefs=None)

Bases: SurfaceProfile

Y Aspheric toroid, even terms up to 20th order, on base conic.

Conics produced for conic constant values:

• cc > 0.0: oblate spheroid

• cc = 0.0: sphere

• cc < 0.0 and > -1.0: ellipsoid

• cc = -1.0: paraboloid

• cc < -1.0: hyperboloid

Conics produced for conic asphere values:

• ec > 1.0: oblate spheroid

• ec = 1.0: sphere

• ec > 0.0 and < 1.0: ellipsoid

• ec = 0.0: paraboloid

• ec < 0.0: hyperboloid

The conic constant is related to the conic asphere as:

• cc = ec - 1

The sag \(z\) is given by:

\(z=f(y)-\frac{1}{2}\textbf{cR}[x^2+z^2-f^2(y)]\)

where \(f(y)=\dfrac{cy^2}{1+\sqrt[](1-\textbf{ec } c^2 y^2)}+\sum_{i=1}^{20} a_iy^{2i}\)

is the sweep profile curve.

initial_size = 10

__init__(c=0.0, cR=0, cc=0.0, r=None, rR=None, ec=None, coefs=None)

initialize a EvenPolynomial profile.

Parameters:

• c – curvature

• r – radius of curvature. If zero, taken as planar. If r is specified, it overrides any input for c (curvature).

• cR – toric sweep radius of curvature

• rR – toric sweep radius

• cc – conic constant

• ec – conic asphere (= cc + 1). If ec is specified, it overrides any input for the conic constant (cc).

• coefs – a list of even power coefficents, starting with the quadratic term, and not exceeding the 20th order term.

property r

property rR

property ec

get_by_order(order: int) → float

Access to coefficients via polynomial order

set_by_order(order: int, value: float)

Access to coefficients via polynomial order

listobj_str()

copyFrom(other)

copyDataFrom(other)

calc_max_nonzero_coef()

apply_scale_factor(scale_factor)

Apply scale_factor to the profile definition.

flip()

update()

sag(x, y)

Returns the sagitta (z coordinate) of the surface at x, y.

fY(y)

f(p)

Returns the value of the profile surface function at point \(\boldsymbol{p}\).

\(f({\boldsymbol{p}}) = 0\)

where \(\boldsymbol{p}=({x, y, z})^T\)

df(p)

Returns the gradient of the profile surface function at point \(\boldsymbol{p}\).

\(df(\boldsymbol{p}) = \partial{f(\boldsymbol{p})}/\partial{\boldsymbol{p}}\)

profile(sd, dir=1, steps=21)

Return a 2d polyline approximating the surface profile.

Parameters:

• sd – semi-diameter of the profile

• dir – +1 for profile from neg to postive direction, -1 if otherwise

• steps – number of points to generate

class XToroid(c=0.0, cR=0, cc=0.0, r=None, rR=None, ec=None, coefs=None)

Bases: YToroid

X Aspheric toroid, even terms up to 20th order, on base conic.

Leverages the YToroid implementation.

Conics produced for conic constant values:

• cc > 0.0: oblate spheroid

• cc = 0.0: sphere

• cc < 0.0 and > -1.0: ellipsoid

• cc = -1.0: paraboloid

• cc < -1.0: hyperboloid

Conics produced for conic asphere values:

• ec > 1.0: oblate spheroid

• ec = 1.0: sphere

• ec > 0.0 and < 1.0: ellipsoid

• ec = 0.0: paraboloid

• ec < 0.0: hyperboloid

The conic constant is related to the conic asphere as:

• cc = ec - 1

The sag \(z\) is given by:

\(z=f(x)-\frac{1}{2}\textbf{cR}[y^2+z^2-f^2(x)]\)

where \(f(x)=\dfrac{cx^2}{1+\sqrt[](1-\textbf{ec } c^2 x^2)}+\sum_{i=1}^{20} a_ix^{2i}\)

is the sweep profile curve.

__init__(c=0.0, cR=0, cc=0.0, r=None, rR=None, ec=None, coefs=None)

initialize a EvenPolynomial profile.

Parameters:

• c – curvature

• r – radius of curvature. If zero, taken as planar. If r is specified, it overrides any input for c (curvature).

• cR – toric sweep radius of curvature

• rR – toric sweep radius

• cc – conic constant

• ec – conic asphere (= cc + 1). If ec is specified, it overrides any input for the conic constant (cc).

• coefs – a list of even power coefficents, starting with the quadratic term, and not exceeding the 20th order term.

sag(x, y)

Returns the sagitta (z coordinate) of the surface at x, y.

f(p)

Returns the value of the profile surface function at point \(\boldsymbol{p}\).

\(f({\boldsymbol{p}}) = 0\)

where \(\boldsymbol{p}=({x, y, z})^T\)

df(p)

Returns the gradient of the profile surface function at point \(\boldsymbol{p}\).

\(df(\boldsymbol{p}) = \partial{f(\boldsymbol{p})}/\partial{\boldsymbol{p}}\)

mutate_profile(old_profile, new_profile_type)

test()